Q: A company wants to estimate the average time (μ) it takes to process an order. They collect a random sample of 36 orders and find a sample mean of 50 minutes. The population standard deviation is known to be 12 minutes. Construct a 95% confidence interval for the population mean processing time. Which of the following is the correct interval?

$$50 \pm 1.96 \times \frac{12}{\sqrt{36}} = (46.08, 53.92)$$

Q: A random sample of 25 hotel rooms in a city has a mean price of $$172.50 and a sample standard deviation of 15.40. $$Assume the population is normal. What is the 95% confidence interval for the true mean price?

$$172.50 \pm 2.0639 \times \frac{15.40}{\sqrt{25}} = (166.14, 178.86)$$

Q: A sample of 100 invoices shows that 10 are in error. Construct a 95% confidence interval for the true proportion of in-error invoices.

$$0.10 \pm 1.96 \times \sqrt{\frac{0.10 \times 0.90}{100}} = (0.0412, 0.1588)$$

Q: A business wants to estimate the average download time for a file within ±5 seconds with 95% confidence. The population standard deviation is 25 seconds. What sample size is required?

$$n = \left( \frac{1.96 \times 25}{5} \$$right)^2$$ = 96.04$$, so 97$

Question 1

A company wants to estimate the average time (μ) it takes to process an order. They collect a random sample of 36 orders and find a sample mean of 50 minutes. The population standard deviation is known to be 12 minutes. Construct a 95% confidence interval for the population mean processing time. Which of the following is the correct interval?

Accepted Answer

$$50 \pm 1.96 	imes \frac{12}{\sqrt{36}} = (46.08, 53.92)$$

Question 2

Which of the following statements best describes the meaning of a 95% confidence interval for a population mean?

Accepted Answer

95% of all confidence intervals constructed from repeated samples will contain the true population mean.

Question 3

A random sample of 25 hotel rooms in a city has a mean price of $$172.50 and a sample standard deviation of 15.40. $$Assume the population is normal. What is the 95% confidence interval for the true mean price?

Accepted Answer

$$172.50 \pm 2.0639 	imes \frac{15.40}{\sqrt{25}} = (166.14, 178.86)$$

Question 4

Which of the following is NOT an assumption for constructing a confidence interval for the mean when the population standard deviation is unknown?

Accepted Answer

The population standard deviation must be known.

Question 5

A sample of 100 invoices shows that 10 are in error. Construct a 95% confidence interval for the true proportion of in-error invoices.

Accepted Answer

$$0.10 \pm 1.96 	imes \sqrt{\frac{0.10 	imes 0.90}{100}} = (0.0412, 0.1588)$$

Question 6

Which of the following increases the width of a confidence interval for the mean, all else held constant?

Accepted Answer

Increasing the confidence level from 95% to 99%.

Question 7

A business wants to estimate the average download time for a file within ±5 seconds with 95% confidence. The population standard deviation is 25 seconds. What sample size is required?

Accepted Answer

$$n = \left( \frac{1.96 	imes 25}{5} \$$right)^2$$ = 96.04$$, so 97$

Question 8

Which of the following is a correct interpretation of a p-value in hypothesis testing?

Accepted Answer

The probability of obtaining a test statistic as extreme or more extreme than the observed value, assuming the null hypothesis is true.

Question 9

A marketing company claims an 8% response rate to its mailings. In a sample of 500, only 25 responses are received. Test the claim at the 0.05 significance level. What is the correct conclusion?

Accepted Answer

Reject the null hypothesis; there is sufficient evidence to reject the company's claim.

Question 10

Which of the following is true about Type I and Type II errors in hypothesis testing?

Accepted Answer

A Type I error is rejecting a true null hypothesis, and a Type II error is failing to reject a false null hypothesis.

Question 11

A sample of 12 has a mean of 112.85 and a standard deviation of 20.8. Test the hypothesis $$H_0$$: \mu = 120$$ vs H_1$: \mu 
eq 120 at$$ $$\alpha = 0.05. $$What is the test statistic?

Accepted Answer

$$t = \frac{112.85 - 120}{20.8/\sqrt{12}} = -1.19$$

Question 12

Which of the following is NOT a requirement for using the normal approximation for a confidence interval for a population proportion?

Accepted Answer

The population must be normally distributed.

Question 13

A manager wants to test if the average monthly cell phone bill is greater than $$52. A sample of 25 bills has a mean of 53.1 $$and a standard deviation of $$10. Test at $$\alpha = 0.10$. What is the correct decision?

Accepted Answer

Do not reject $$H_0$$; there is not sufficient evidence that the mean bill is over $52.

Question 14

Which of the following best describes the relationship between confidence intervals and two-tailed hypothesis tests?

Accepted Answer

If the hypothesized value is outside the confidence interval, the null hypothesis is rejected at the corresponding significance level.

Question 15

Which of the following factors will decrease the probability of a Type II error (β) in hypothesis testing?

Accepted Answer

Increasing the sample size.

Question 16

A researcher wants to estimate the proportion of customers who prefer a new product. She wants the estimate to be within ±0.07 with 95% confidence. A pilot sample gives $$p = 0.15. $$What is the minimum sample size required?

Accepted Answer

$$n = \left( \frac{1.96^2$ 	imes 0.15 	imes 0.85}{0.07^2$} ight) \approx 100$$

Question 17

Which of the following is an ethical requirement when reporting the results of a statistical analysis?

Accepted Answer

Reporting both statistically significant and non-significant results.

Confidence Intervals and Hypothesis Testing: Study Notes for Business Statistics

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